Statistics — Study Notes

1. Introduction

Statistics is the science of collecting, organising, summarising, and interpreting data. When data sets are large, it is useful to describe them with a few key numbers that capture their central tendency and their spread. This chapter extends the measures of central tendency to a more complete picture by introducing measures of dispersion.

The chapter begins by recalling the mean, median, and mode, and then focuses on the measures of dispersion: range, mean deviation, variance, and standard deviation. It develops formulas for grouped and ungrouped data, for both individual observations and frequency distributions, and studies the variance and standard deviation of grouped data using the assumed mean method and step deviation method.

Measures of dispersion answer a critical question that averages cannot: how spread out is the data? Two data sets can have the same mean but very different variability, which matters enormously in quality control, finance, weather, and science. Understanding variance and standard deviation is also the foundation of probability distributions and inferential statistics studied later.

2. Measures of Central Tendency

The measures of central tendency give a single representative value of a data set.

Mean

The arithmetic mean of n observations x1, x2, ..., xn is:

x bar = (x1 + x2 + ... + xn)/n

For a frequency distribution with values xi and frequencies fi, the mean is:

x bar = (sum fi xi)/(sum fi)

Median

The median is the middle value when the observations are arranged in increasing order. For n observations:

• If n is odd, the median is the ((n + 1)/2)-th observation.
• If n is even, the median is the average of the (n/2)-th and (n/2 + 1)-th observations.

Mode

The mode is the observation that occurs most frequently.

3. Measures of Dispersion

Dispersion measures how scattered the data values are around the central value.

Range

The range is the difference between the maximum and minimum values:

Range = Maximum - Minimum

The range is easy to compute but uses only two values, ignoring the rest of the data.

Mean Deviation

The mean deviation about a central point A is the mean of the absolute deviations:

MD(A) = (sum |xi - A|)/n

For grouped data with frequencies:

MD(A) = (sum fi |xi - A|)/(sum fi)

The central point A is usually the mean or the median. Mean deviation is least when taken about the median.

Variance and Standard Deviation

The variance is the mean of the squares of the deviations from the mean. For ungrouped data:

sigma^2 = sum (xi - x bar)^2 / n

The standard deviation is the square root of the variance:

sigma = sqrt(sum (xi - x bar)^2 / n)

The standard deviation has the same units as the data, making it easier to interpret.

4. Variance and Standard Deviation of Grouped Data

For a Frequency Distribution

sigma^2 = sum fi (xi - x bar)^2 / N, where N = sum fi

Shortcut Formulas

Using the shortcut formula, the variance can be computed without finding deviations individually:

sigma^2 = (sum fi xi^2)/N - (x bar)^2

For grouped data with class marks xi:

sigma^2 = (sum fi (xi - x bar)^2)/N

The standard deviation is sqrt of the variance.

Methods of Computation

For grouped data, three methods are commonly used:

1. Direct method: compute x bar and then the squared deviations.
2. Assumed mean method: use a guessed mean to simplify arithmetic.
3. Step deviation method: divide deviations by the class width h to make numbers smaller.

In the step deviation method, define ui = (xi - A)/h, then:

x bar = A + (sum fi ui)/N x h

sigma = h sqrt((sum fi ui^2)/N - ((sum fi ui)/N)^2)

5. Coefficient of Variation

The coefficient of variation (CV) is a relative measure of dispersion:

CV = (sigma/x bar) x 100%

It expresses the standard deviation as a percentage of the mean. The CV is used to compare the variability of two data sets that may have different means or units. The group with the smaller coefficient of variation is more consistent or stable.

6. Comparing Two Data Sets

Two data sets can be compared using their means and standard deviations:

• The group with the higher mean has the higher average value.
• The group with the lower coefficient of variation is more consistent.

For example, if two manufacturers produce bulbs with the same mean lifetime but different standard deviations, the one with the smaller standard deviation is more reliable.

Quick Revision Tables

Table 1: Measures of Dispersion for Ungrouped Data

Table 2: Coefficient of Variation and Grouped Data

Mind Map

Important Diagrams (SVG)

Diagram 1: Mean, Median and Mode of a Distribution

Diagram 2: Variance as Average Squared Deviation

Common Mistakes

1. Using the range as a reliable measure of dispersion; it ignores all data except the maximum and minimum.
2. Computing variance as the mean of deviations without squaring. The deviations sum to zero, so squaring is essential.
3. Forgetting to take the square root at the end. Variance and standard deviation are different quantities.
4. Using the ungrouped formula for grouped data. For frequency distributions, use sum fi (xi - x bar)^2 / N.
5. Confusing the population formula with the sample formula; this chapter uses the population-based formulas with division by N.
6. In the shortcut formula, forgetting to subtract (x bar)^2 from (sum xi^2)/N.
7. Saying mean deviation is least about the mean. It is least about the median.
8. Interpreting a larger standard deviation as more consistent. A larger standard deviation means more spread, hence less consistency.
9. Forgetting to multiply by h in the step deviation method when converting ui back to xi.

Exam Tips

1. For ungrouped data, compute the mean first, then the deviations, squares, and average to find variance.
2. For grouped data, always verify that the class marks are used as xi in the formulas.
3. When comparing consistency, always use the coefficient of variation, not the standard deviation alone, especially when means differ.
4. Use the shortcut formula sigma^2 = (sum xi^2)/n - (xbar)^2 to save time on ungrouped data.
5. In the step deviation method, define ui = (xi - A)/h clearly and remember the mean is A + h(sum fi ui)/N.
6. Practise converting between variance and standard deviation quickly; the standard deviation is always the positive square root.
7. For the median, sort the data first and apply the n odd/even rule correctly.

Conclusion

Statistics provides the tools to summarise data and understand its variability. Measures of central tendency tell us where the data is centred, while measures of dispersion such as the range, mean deviation, variance, and standard deviation tell us how the data is spread. The standard deviation, in particular, is a fundamental measure that appears throughout probability and statistics. The coefficient of variation enables fair comparisons between data sets with different scales or means. These methods are essential in science, business, and everyday decision-making, and they prepare students for the study of probability distributions and statistical inference in higher classes.